Proves the Erdős–Faudree–Rousseau–Schelp conjecture: R(C_m,K_n)=(m-1)(n-1)+1 for every m\ge n\ge3, except R(C_3,K_3)=6. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph. (Lean)
Manuscript
Cycle--clique Ramsey numbers
September 25, 2026 · source · cite as \cite{OAI:Cycle-clique-Ramsey-numbers-September-25-2026}
We prove that R(C_m,K_n)=(m-1)(n-1)+1 for every pair of integers m\ge n\ge3 other than (m,n)=(3,3), for which R(C_3,K_3)=6. This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. These arguments reduce the remaining cases to 3{,}099 finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules. Complete programs and deduction traces accompany the paper.
Result 189 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md